The rank condition and strong rank conditions for Ore extensions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908907565219840 |
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| author | Lorensen, Karl Öinert, Johan |
| author_facet | Lorensen, Karl Öinert, Johan |
| contents | Let $R$ be a ring, $σ:R\to R$ a ring endomorphism, and $δ:R\to R$ a $σ$-derivation. We establish that the Ore extension $R[x;σ,δ]$ satisfies the rank condition if and only if $R$ does. In addition, we prove analogous results for the right and left strong rank conditions. However, in the right case, the ``if" part requires the hypothesis that $σ$ is an automorphism, whereas, in the left case, this assumption is needed for the ``only if" part. Finally, we provide a new proof of an old result of Susan Montgomery stating that a skew power series ring is directly (respectively, stably) finite if and only if its coefficient ring is directly (respectively, stably) finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21030 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The rank condition and strong rank conditions for Ore extensions Lorensen, Karl Öinert, Johan Rings and Algebras 16P99, 16S32, 16S36, 16S99, 16W25, 16W70 Let $R$ be a ring, $σ:R\to R$ a ring endomorphism, and $δ:R\to R$ a $σ$-derivation. We establish that the Ore extension $R[x;σ,δ]$ satisfies the rank condition if and only if $R$ does. In addition, we prove analogous results for the right and left strong rank conditions. However, in the right case, the ``if" part requires the hypothesis that $σ$ is an automorphism, whereas, in the left case, this assumption is needed for the ``only if" part. Finally, we provide a new proof of an old result of Susan Montgomery stating that a skew power series ring is directly (respectively, stably) finite if and only if its coefficient ring is directly (respectively, stably) finite. |
| title | The rank condition and strong rank conditions for Ore extensions |
| topic | Rings and Algebras 16P99, 16S32, 16S36, 16S99, 16W25, 16W70 |
| url | https://arxiv.org/abs/2505.21030 |